An interval observer for uncertain continuous‐time linear systems

To design robust interval observers for uncertain continuous‐time linear systems, a new set‐integration approach is proposed to compute trajectory tubes for the estimation error. Because this approach, the order‐preserving condition on the dynamics of the estimation error is no longer required. Therefore, synthesis methods can be used to compute observer gains that reduce the impact of the system uncertainties on the accuracy of the estimated state enclosures. The performance of the proposed approach is showcased through illustrative numerical examples.     

Interactive real-time physics: An intuitive approach to form-finding and structural analysis for design and education

Real-time physics simulation has been extensively used in computer games, but its potential has yet to be fully realized in design and education. We present an interactive 3D physics engine with a wide variety of applications.In common with traditional FEM, the use of a local element stiffness matrix is retained. However, unlike typical non-linear FEM routines elements forces, moments and inertia are appropriately lumped at nodes following the dynamic relaxation method. A semi-implicit time integration scheme updates linear and angular momentum, and subsequently the local coordinate frames of the nodes. A co-rotational approach is used to compute the resultant field of displacements in global coordinates including the effect of large deformations. The results obtained compare well against established commercial software.We demonstrate that the method presented allows the making of interactive structural models that can be used in teaching to develop an intuitive understanding of structural behaviour. We also show that the same interactive physics framework allows real-time optimization that can be used for geometric and structural design applications.     

A parallel aerostructural optimization framework for aircraft design studies

Preliminary aircraft design studies use structural weight models that are calibrated with data from existing aircraft. Computing weights with these models is a fast procedure that provides reliable weight estimates when the candidate designs lie within the domain of the data used for calibration. However, this limitation is too restrictive when we wish to assess the relative benefits of new structural technologies and new aircraft configurations early in the design process. To address this limitation, we present a computationally efficient aerostructural design framework for initial aircraft design studies that uses a full finite-element model of key structural components to better assess the potential benefits of new technologies. We use a three-dimensional panel method to predict the aerodynamic forces and couple the lifting surface deflections to compute the deformed aerodynamic flying shape. To be used early in the design cycle, the aerostructural computations must be fast, robust, and allow for significant design flexibility. To address these requirements, we develop a geometry parametrization technique that enables large geometric modifications, we implement a parallel Newton–Krylov approach that is robust and computationally efficient to solve the aeroelastic system, and we develop an adjoint-based derivative evaluation method to compute the derivatives of functions of interest for design optimization. To demonstrate the capabilities of the framework, we present a design optimization of a large transport aircraft wing that includes a detailed structural design parametrization. The results demonstrate that the proposed framework can be used to make detailed structural design decisions to meet overall aircraft mission requirements.     

Bent-beam electrothermal actuators-Part II: Linear and rotarymicroengines

For Part I see L. Que, J.S. Park and Y.B. Gianchandani, ibid., vol.10, pp.247-54 (2001). This paper reports on the use of bent-beam electrothermal actuators for the purpose of generating rotary and long-throw rectilinear displacements. The rotary displacements are achieved by orthogonally arranged pairs of cascaded actuators that are used to rotate a gear. Devices were fabricated using electroplated Ni, p ⁺⁺ Si, and polysilicon as structural materials. Displacements of 20-30 μm with loading forces >150 μN at actuation voltages <12 V and power dissipation <300 mW could be achieved in the orthogonally arranged actuator pairs. A design that occupies <1 mm ² area is presented. Long-throw rectilinear displacements were achieved by inchworm mechanisms in which pairs of opposing actuators grip and shift a central shank that is cantilevered on a flexible suspension. A passive lock holds the displaced shank between pushes and when the power is off. This arrangement permits large output forces to be developed at large displacements, and requires zero standby power. Several designs were fabricated using electroplated Ni as the structural material. Forces >200 μN at displacements >100 μm were measured     

Cable stretching force optimization of concrete cable-stayed bridges including construction stages and time-dependent effects

This paper presents an optimization algorithm to compute the prestressing forces on concrete cable-stayed bridges to achieve the desired final geometry. The structural analysis includes the load history and geometry changes due to the construction sequence and the time-dependent effects due to creep, shrinkage and aging of the concrete. An entropy-based approach was used for structural optimization and discrete direct sensitivity analysis was used to evaluate the structural response to changes in the design variables. Numerical examples are presented and the results exhibit the importance of considering both the construction stages and the time-dependent effects for adequately predict the bridge behaviour and compute the cable prestressing forces.     

Interval Finite Elements as a Basis for Generalized Models of Uncertainty in Engineering Mechanics

Latest scientific and engineering advances have started to recognize the need for defining multiple types of uncertainty. Probabilistic modeling cannot handle situations with incomplete or little information on which to evaluate a probability, or when that information is nonspecific, ambiguous, or conflicting [12], [47], [50]. Many interval-based uncertainty models have been developed to treat such situations. This paper presents an interval approach for the treatment of parameter uncertainty for linear static structural mechanics problems. Uncertain parameters are introduced in the form of unknown but bounded quantities (intervals). Interval analysis is applied to the Finite Element Method (FEM) to analyze the system response due to uncertain stiffness and loading. To avoid overestimation, the formulation is based on an element-by-element (EBE) technique. Element matrices are formulated, based on the physics of materials, and the Lagrange multiplier method is applied to impose the necessary constraints for compatibility and equilibrium. Earlier EBE formulation provided sharp bounds only on displacements [32]. Based on the developed formulation, the bounds on the system’s displacements and element forces are obtained simultaneously and have the same level of accuracy. Very sharp enclosures for the exact system responses are obtained. A number of numerical examples are introduced, and scalability is illustrated.     

Extended interval Newton method based on the precise quotient set

The interval Newton method can be used for computing an enclosure of a single simple zero of a smooth function in an interval domain. It can practically be extended to allow computing enclosures of all zeros in a given interval. This paper deals with the extended interval Newton method. An essential operation of the method is division by an interval that contains zero (extended interval division). This operation has been studied by many researchers in recent decades, but inconsistency in the research has occurred again and again. This paper adopts the definition of extended interval division redefined in recent documents (Kulisch in Arithmetic operations for floating-point intervals, 2009; Pryce in P1788: IEEE standard for interval arithmetic version 02.2, 2010). The result of the division is called the precise quotient set. Earlier definitions differ in the overestimation of the quotient set in particular cases, causing inefficiency in Newton’s method and even leading to redundant enclosures of a zero. The paper reviews and compares some extended interval quotient sets defined during the last few decades. As a central theorem, we present the fundamental properties of the extended interval Newton method based on the precise quotient set. On this basis, we develop an algorithm and a convenient program package for the extended interval Newton method. Statements on its convergence are also given. We then demonstrate the performance of the algorithm through nine carefully selected very sensitive numerical examples and show that it can compute correct enclosures of all zeros of the functions with high efficiency, particularly in cases where earlier methods are less effective.     

Interval arithmetic techniques for the design of controllers for nonlinear dynamical systems with applications in mechatronics. II

In this paper, we give an overview of interval arithmetic techniques for both the offline and online verification of robust control and observer strategies. In Part 1 of this paper, we have introduced basic interval arithmetic techniques focusing on offline applications while we deal with their online application in this Part 2. In contrast to the offline tasks considered in Part 1, in which guaranteed enclosures of all admissible solutions to a specific control problem are computed, real-time requirements have to be considered in online tasks. This means that we have to ensure to find a guaranteed admissible solution within a limited computing time. Therefore, the adapted goal in the online design of controllers and observers must be to determine at least one solution, and not necessarily all, such that feasibility under the influence of uncertainties is proven. After a summary of the basic computational procedures, first experimental results for the real-time application of interval tools for feedforward control and disturbance estimation are presented for the temperature control of a distributed heating system.     

Ray tracing general parametric surfaces using interval arithmetic

This paper describes an algorithm for ray tracing general parametric surfaces. After dividing the surface adaptively into small parts, a binary tree of these parts is built. For each part a bounding volume is calculated with interval arithmetic. From linear approximations and intervals for the partial derivatives it is possible to construct parallelepipds that adapt the orientation and shape of the surface parts very well and form very tight enclosures. Therefore we can develop an algorithm for rendering that is similar to that used with Bèzier and B-spline surfaces, where the bounding volumes are derived from the convex hull property. The tree of enclosures (generated once in a preprocessing step) guarantees that each ray that hits the surface leads to an iteration on a very small surface part; this iteration can be robustly (and very quickly) performed in real arithmetic.     

A combined method for enclosing all solutions of nonlinear systems of polynomial equations

We consider the problem of finding interval enclosures of all zeros of a nonlinear system of polynomial equations. We present a method which combines the method of Gröbner bases (used as a preprocessing step), some techniques from interval analysis, and a special version of the algorithm of E. Hansen for solving nonlinear equations in one variable. The latter is applied to a triangular form of the system of equations, which is generated by the preprocessing step. Our method is able to check if the given system has a finite number of zeros and to compute verfied enclosures for all these zeros. Several test results demonstrate that our method is much faster than the application of Hansen’s multidimensional algorithm (or similar methods) to the original nonlinear systems of polynomial equations.     

Structural analysis of francis turbine runners using ADINA

ADINA is used as the primary design tool in the structural analysis of Francis turbine runners at Kværner Energy. An in-house developed finite-element mesh generator is used for quick and efficient modelling of any runner or runner vane segment in THREEDSOLID elements. The finite-element analyses include frequency analyses in order to avoid structural amplification of dynamic loads and static-strength analyses in order to obtain a uniform stress distribution in the runner. Comparison of the convergence has been carried out with respect to mesh density and order of the elements. The high-order elements (20- and 27-nodes) compute both smaller displacements and lower natural frequencies in comparison to eight-node elements. For stress calculations, only high-order elements are used. Poor structural design is one explanation of failures in some runners. Knowledge of structural performance secures structural integrity and reveals the potential for reducing weight.     

Analysis of validated error bounds of surface-to-surface intersection

In this paper, we address the problem of computing the error bounds of surface-to-surface intersection and propose a novel procedure to reduce them. We formulate the surface-to-surface intersection problem as solving a system of ordinary differential equations and using the validated ODE solver we compute the validated a priori enclosures of an intersection curve, in which the existence and the uniqueness of a solution are guaranteed. Then we use straight line enclosures to reduce the size of the a priori enclosures. These reduced enclosures are again enclosed by bounding curves, which can be used as the reduced error bounds of the intersection curve. We demonstrate our method with tangential and transversal intersections.     

Dynamic response analyses of trusses under harmonic loading

The stiffness method of analysis is used to compute displacements and end-bar forces for pinconnected trusses under the action of harmonic forces. Three different formulations are presented to account for the inertia forces: (1) the ‘exact’ analysis based on the Bernoulli-Euler equation, (2) the series expansion approximation, and (3) the lumped mass method. A computer program is provided with options to perform the analysis by any of these three methods.     

Seismic stability of cracked concrete dams using rigid block models

Several gravity dams subjected to severe ground motions are likely to experience cracking and sliding in the upper section where dynamic amplification is important. A high acceleration spike realistically applies large inertia forces computed from mass times the acceleration. However, these impulsive inertia forces might not be applied in the same direction for a sufficient long period of time to induce significant rotational or sliding displacements detrimental to the seismic or post-seismic structural stability of the cracked components. When it is of interest to estimate residual sliding displacements, a convenient and simple tool is to perform transient rigid body “sliding block” analysis of the “cracked” component. However, this requires the definition of proper seismic input motions at the base of the block with due consideration of dynamic amplification. The possibility to compute in-structure response spectra (ISRS) at the base of the block to define suitable spectra compatible accelerograms is presented in this paper. An important conclusion is that it is not conservative to use accelerograms compatible with the linear (uncracked) dam ISRS to perform transient rigid body sliding response analyses. Dam base and upper joint cracking affects its dynamic properties such that there are modifications of the intensities and frequency content of seismic motions as they propagate over the dam’s height. An envelope of nonlinear ISRS computed from cracked beam models of the dam is recommended to obtain compatible accelerograms and to provide a conservative estimate of upper block residual sliding displacements.     

Enhancing numerical constraint propagation using multiple inclusion representations

Building tight and conservative enclosures of the solution set is of crucial importance in the design of efficient complete solvers for numerical constraint satisfaction problems (NCSPs). This paper proposes a novel generic algorithm enabling the cooperative use, during constraint propagation, of multiple enclosure techniques. The new algorithm brings into the constraint propagation framework the strength of techniques coming from different areas such as interval arithmetic, affine arithmetic, and mathematical programming. It is based on the directed acyclic graph (DAG) representation of NCSPs whose flexibility and expressiveness facilitates the design of fine-grained combination strategies for general factorable systems. The paper presents several possible combination strategies for creating practical instances of the generic algorithm. The experiments reported on a particular instance using interval constraint propagation, interval arithmetic, affine arithmetic, and linear programming illustrate the flexibility and efficiency of the approach.     

A numerical strategy to compute optical parameters in turbulent flow: Application to telescopes

We present a numerical formulation to compute optical parameters in a turbulent air flow. The basic numerical formulation is a large eddy simulation (LES) of the incompressible Navier-Stokes equations, which are approximated using a finite element method. From the time evolution of the flow parameters we describe how to compute statistics of the flow variables and, from them, the parameters that determine the quality of the visibility. The methodology is applied to estimate the optical quality around telescope enclosures.     

Uncertainty analysis of structural systems by perturbation techniques

The formulation of an efficient method to evaluate the uncertainty of the structural response by applying perturbation techniques is described. Structural random variables are defined by their mean values, standard deviations and correlations. The uncertainty of structural behaviour is evaluated by the covariance matrix of response according to the developed perturbation methodology. It is also presented the procedure used to implement this method in a structural finite element framework. The implemented computational program allows, in only one structural analysis, to evaluate the mean value and the standard deviation of the structural response, defined in terms of displacements or forces. The proposed method is exact for problems with linear design functions and normal-distributed random variables. Results remain accurate for non-linear design functions if they can be approximated by a linear combination of the basic random variables.     

Fuzzy sets in seismic inelastic analysis and design of reinforced concrete frames

In this paper, a new approach to the problem of estimating the structural response of systems with uncertain characteristics is presented. The approach is based on the theory of fuzzy sets, which allow the designers to describe the uncertain variables. The method is presented briefly in the following. First, the uncertain parameters are expressed as fuzzy numbers with specific characteristics. The concurrent effect of the various uncertainties on the structural response is obtained by applying methodologies of the theory of fuzzy sets. Then the output parameters of the design process as, e.g. the displacements or the stresses of the structure are obtained as new fuzzy numbers expressing the uncertainties of the output parameters. Finally, numerical applications on a number of relatively simple structural systems give an idea of the applicability of the proposed methodology in various aspects of the design process.     

Clouds,Fuzzy Sets,and Probability Intervals

Clouds are a concept for uncertainty mediating between the concept of a fuzzy set and that of a probability distribution. A cloud is to a random variable more or less what an interval is to a number. We discuss the basic theoretical and numerical properties of clouds, and relate them to histograms, cumulative distribution functions, and likelihood ratios.We show how to compute nonlinear transformations of clouds, using global optimization and constraint satisfaction techniques. We also show how to compute rigorous enclosures for the expectation of arbitrary functions of random variables, and for probabilities of arbitrary statements involving random variables, even for problems involving more than a few variables.Finally, we relate clouds to concepts from fuzzy set theory, in particular to the consistent possibility and necessity measures of Jamison and Lodwick.     

Aeroelastic tailoring considering uncertainties in material properties

The robustness of aeroelastic design optimization with respect to uncertainties in material and structural properties is studied both numerically and experimentally. The model consists of thin orthotropic composite wings virtually without fuselage. Three different configurations with consistent geometry but varying orientation of the main stiffness axis of the material are investigated. The onset of aeroelastic instability, flutter, is predicted using finite element analysis and the doublet-lattice method for the unsteady aerodynamic forces. The numerical results are experimentally verified in a low-speed wind tunnel. The optimization problem is stated as to increase the critical air speed, above that of the bare wing by massbalancing. It is seen that the design goals are not met in the experiments due to uncertainties in the structural performance of the wings. The uncertainty in structural performance is quantified through numerous dynamic material tests. Once accounting for the uncertainties through a suggested reformulation of the optimization problem, the design goals are met also in practice. The investigation indicates that robust and reliable aeroelastic design optimization is achievable, but careful formulation of the optimization problem is essential.